It doesn’t take too much delving around with the primes to come across the appearance of Euler’s totient or phi function. It oozes out of the primes in a variety of locations.
Firstly the composite helices and thus the prime helices each have a “width”. The width of the 6n+/-1 helix associated with prime factor 5 is 2. The width of the next helix associated with prime factor 7 is 8. The next is 48 and the conjecture is that this continues:
Secondly, considering the balance of primes to composites as each prime factor is used to calculate which are composite and which are potentially prime we see a simple pattern:
Each block is pi# integers long,
Each block has (pi – 1) # (potential) primes & (pi-1 -1)# composites
Thus the ratio of
primes to all integers: (pi – 1) # / pi#
primes to composites (pi – 1) # / (pi-1 -1) # = pi – 1